A double Mersenne number is a number of the form
 |
where 
is a Mersenne number. The first few double Mersenne
numbers are 1, 7, 127, 32767, 2147483647, 9223372036854775807, ... (OEIS A077585).
A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne prime 
can be prime only for prime 
, a double Mersenne prime can be prime only for prime 
, i.e., 
a Mersenne prime. Double
Mersenne numbers are prime for 
, 3, 5, 7, corresponding to the sequence 7, 127, 2147483647,
170141183460469231731687303715884105727, ... (OEIS A077586).
The next four 
,
,
,
and 
have known factors summarized in the following table. The status of all other double
Mersenne numbers is unknown, with 
being the smallest unresolved case. Since this number
has 694127911065419642 digits, it is much too large for the usual Lucas-Lehmer
test to be practical. The only possible method of determining the status of this
number is therefore attempting to find small divisors (or discovery of an efficient
primality test for this type of number). T. Forbes
has organized a distributed search, but thus no factors have been found although
about 80% of the trial divisors up to 
have been checked. Edgington maintains
a list of known factorizations of double Mersenne numbers.
 | factors | reference |
| 13 | 338193759479, C2455 | Wilfrid Keller (1976) |
| 17 | 231733529 | Raphael Robinson (1957) |
| 19 | 62914441 | Raphael Robinson (1957) |
| 31 | 295257526626031 | Guy Haworth (1983, 1987) |
| 87054709261955177 | Keller (1994) | |
| 242557615644693265201 | Keiser and Forbes (1999) | |
| 178021379228511215367151 | Mayer (2005) |
." 
. Progress: 2 March 2004."